Final answer:
To derive the slope when α0 is zero, the equation becomes y = bx, implying the line passes through the origin. If 20 times the slope b is zero, the equation represents a horizontal line at y = α0, indicating no change in y with x.
Step-by-step explanation:
The student's question involves understanding the terms of the linear regression equation y = α0 + bx + ε. In this context, α0 (alpha-zero) represents the y-intercept, while b represents the slope of the regression line. The variable ε is the error term.
(a) The student asks to derive the slope term when α0 equals zero. If α0 is zero, then the regression line passes through the origin (x,y) = (0,0). Without altering the slope b, the equation simplifies to y = bx.
(b) The student also asks to derive what happens when 20 times the slope term (b = 0) is zero. If b is zero, the slope is horizontal, indicating there's no change in y as x increases. Thus, the regression line becomes a horizontal line at the level of the y-intercept, expressed as y = α0.
It's important to note that the interplay of the slope and y-intercept determines the overall shape of the regression line, with slope dictating the steepness and direction, and the y-intercept indicating the starting point of the line on the y-axis.